Page 14 - Computational Fluid Dynamics for Engineers

Contents XIII

10. Inviscid Compressible Flow 295
10.1 Introduction 295
10.2 Shock Jump Relations 296
10.3 Shock Capturing 299
10.4 The Transonic Small Disturbance (TSD) Equation 301
10.5 Model Problem for the Transonic Small Disturbance Equation:
Flow Over a Non-Lifting Airfoil 302
10.5.1 Discretized Equation 303
10.5.2 Solution Procedure and Sample Calculations 304
10.6 Solution of Full-Potential Equation 308
10.7 Boundary Conditions for the Euler Equations 309
10.8 Stability Analysis of the Euler Equations 311
10.9 MacCormack Method for Compressible Euler Equations 312
10.10 Model Problem for the MacCormack Method:
Unsteady Shock Tube 313
10.10.1 Initial Conditions 314
10.10.2 Boundary Conditions 314
10.10.3 Solution Procedure and Sample Calculations 314
10.11 Model Problem for the MacCormack Method:
Quasi 1-D Nozzle 315
10.11.1 Initial Conditions 316
10.11.2 Boundary Conditions 317
10.11.3 Solution Procedure and Sample Calculations 318
10.12 Beam-Warming Method for Compressible Euler Equations . . . . 320
10.13 Model Problem for the Implicit Method: Unsteady Shock Tube 321
10.13.1 Solution Procedure and Sample Calculations 321
10.14 Model Problem for the Implicit Method: Quasi-ID Nozzle 322
10.14.1 Solution Procedure and Sample Calculations 325
References 326
Problems 326

11. Incompressible Navier-Stokes Equations 327
11.1 Introduction 327
11.2 Analysis of the Incompressible Navier-Stokes Equations 328
11.3 Boundary Conditions 329
11.4 Artificial Compressibility Method: INS2D 331
11.4.1 Discretization of the Artificial Time Derivatives 331
11.4.2 Discretization of the Convective Fluxes 332
11.4.3 Discretization of the Viscous Fluxes 334
11.4.4 System of Discretized Equation 335
11.5 Model Problem: Sudden Expansion Laminar Duct Flow 336
11.5.1 Discretization of the Boundary Conditions 337